Univalence of horizontal shear of Ces\`aro type transforms

Abstract

This manuscript investigates the classical problem of determining conditions on the parameters $\alpha,\beta \in \mathbb{C}$ for which the integral transform $$C_{\alpha\beta}[\varphi](z):=\int_{0}^{z} \bigg(\frac{\varphi(\zeta)}{\zeta (1-\zeta)^{\beta}}\bigg)^\alpha\,d\zeta $$ is also univalent in the unit disk, where $\varphi$ is a normalized univalent function. Additionally, whenever $\varphi$ belongs to some subclasses of the class of univalent functions, the univalence features of the harmonic mappings corresponding to $C_{\alpha\beta}[\varphi]$ and its rotations are derived. As applications to our primary findings, a few non-trivial univalent harmonic mappings are also provided. The primary tools employed in this manuscript are Becker's univalence criteria and the shear construction developed by Clunie and Sheil-Small.